Skip to main content
SpringerLink
Log in
Book cover

Modeling in Mathematics pp 117–127Cite as

Avoiding Higher Matrix Powers in the Solution of Linear Dynamical Systems

  • Conference paper
  • First Online:

  • 567 Accesses

Part of the book series: Atlantis Transactions in Geometry ((ATLANTIS,volume 2))

Abstract

We show that, using Lucas polynomials of the second kind, it is possible to write down explicitly the solution of linear dynamical systems - both in the discrete and continuous case - avoiding higher matrix powers. This improves the computational complexity of the algorithms usually described in literature.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   149.00
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. M.W.  Hirsch, S.  Smale, R.L.  Devaney, Differential equations, Dynamical Systems and An Introduction to Chaos (Academic Press, (Elsevier), San Diego - London, 2003)

    Google Scholar 

  2. E.R. Scheinerman, Invitation to Dynamical Systems (Dover, New York, 2012)

    MATH  Google Scholar 

  3. É. Lucas, Théorie des Nombres (Gauthier-Villars, Paris, 1891)

    MATH  Google Scholar 

  4. I.V.V. Raghavacharyulu, A.R. Tekumalla, Solution of the Difference Equations of Generalized Lucas Polynomials. J. Math. Phys. 13, 321–324 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Bruschi, P.E. Ricci, An explicit formula for \(f({\cal{A}})\) and the generating function of the generalized Lucas polynomials. Siam J. Math. Anal. 13, 162–165 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. M.  Bruschi, P.E.  Ricci, I polinomi di Lucas e di Tchebycheff in più variabili, Rend. Mat., S. VI, 13, (1980), 507–530

    Google Scholar 

  7. P.E.  Ricci, Sulle potenze di una matrice, Rend. Mat. (6) 9 (1976), 179–194

    Google Scholar 

  8. F.R. Gantmacher, Matrix Theory (Chelsea Pub. Co, New York, 1960)

    Google Scholar 

  9. R. Lidl, C. Wells, Chebyshev polynomials in several variables. J. Reine Angew. Math. 255, 104–111 (1972)

    MathSciNet  MATH  Google Scholar 

  10. R. Lidl, Tschebyscheff polynome in mehreren variabelen. J. Reine Angew. Math. 273, 178–198 (1975)

    MathSciNet  MATH  Google Scholar 

  11. T.H.  Koornwinder, Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators, I–II. Kon. Ned. Akad. Wet. Ser. A, 77, 46–66 (1974)

    Google Scholar 

  12. T.H. Koornwinder, Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators, III-IV. Indag. Math. 36, 357–381 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  13. K.B. Dunn, R. Lidl, Multi-dimensional generalizations of the Chebyshev polynomials, I-II. Proc. Japan Acad. 56, 154–165 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. R.J. Beerends, Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator. Trans. Am. Math. Soc. 328(2), 779–814 (1991)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Physics, Roma Tre University, Largo San Leonardo Murialdo, 1, 00146, Rome, Italy

    Pierpaolo Natalini

  2. International Telematic University UniNettuno, Corso Vittorio Emanuele II, 39, 00186, Rome, Italy

    Paolo Emilio Ricci

Authors
  1. Pierpaolo Natalini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Paolo Emilio Ricci
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierpaolo Natalini .

Editor information

Editors and Affiliations

  1. Bio-Engineering Sciences, University of Antwerp, Antwerp, Belgium

    Johan Gielis

  2. International Telematic University UniNettuno, Roma, Italy

    Paolo Emilio Ricci

  3. Faculty of Exact and Natural Sciences, Ivane Javakhishvili Tbilisi State University, Tbilisi, Georgia

    Ilia Tavkhelidze

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Atlantis Press and the author(s)

About this paper

Cite this paper

Natalini, P., Ricci, P.E. (2017). Avoiding Higher Matrix Powers in the Solution of Linear Dynamical Systems. In: Gielis, J., Ricci , P., Tavkhelidze, I. (eds) Modeling in Mathematics . Atlantis Transactions in Geometry, vol 2. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-261-8_9

Download citation

Publish with us

Policies and ethics

Search

Navigation